{"paper":{"title":"The Randi\\'{c} index and signless Laplacian spectral radius of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo Ning, Xing Peng","submitted_at":"2016-01-14T07:48:36Z","abstract_excerpt":"Given a connected graph $G$, the Randi\\'c index $R(G)$ is the sum of $\\tfrac{1}{\\sqrt{d(u)d(v)}}$ over all edges $\\{u,v\\}$ of $G$, where $d(u)$ and $d(v)$ are the degree of vertices $u$ and $v$ respectively. Let $q(G)$ be the largest eigenvalue of the singless Laplacian matrix of $G$ and $n=|V(G)|$. Hansen and Lucas (2010) made the following conjecture: \\[ \\frac{q(G)}{R(G)} \\leq \\begin{cases}\n  \\frac{4n-4}{n} & 4 \\leq n\\leq 12\n  \\frac{n}{\\sqrt{n-1}} & n\\geq 13\n  \\end{cases} \\] with equality if and only if $G=K_{n}$ for $4\\leq n\\leq 12$ and $G=S_n$ for $n\\geq 13$, respectively. Deng, Balachandr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03511","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}